A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of $B_n$

TL;DR

This paper constructs Dubrovin-Frobenius manifold structures on the orbit space of the group $B_n$, linking them to NLS equations and extending previous $B_2$ results to arbitrary $n$, with connections to KP equations.

Contribution

It generalizes the Dubrovin-Frobenius manifold structure from $B_2$ to arbitrary $B_n$, relating it to NLS equations and invariant bilinear forms.

Findings

Constructed two Dubrovin-Frobenius structures for $B_2$ orbit space.

Extended the defocusing case to all $B_n$, establishing a new geometric structure.

Prepotentials match those of constrained KP equations up to $n=4$.

Abstract

Generalizing a construction presented in [3], we show that the orbit space of $B_2$ less the image of coordinate lines under the quotient map is equipped with two Dubrovin-Frobenius manifold structures which are related respectively to the defocusing and the focusing nonlinear Schr\"odinger (NLS) equations. Motivated by this example, we study the case of $B_n$ and we show that the defocusing case can be generalized to arbitrary $n$ leading to a Dubrovin-Frobenius manifold structure on the orbit space of the group. The construction is based on the existence of a non-degenerate and non-constant invariant bilinear form that plays the role of the Euclidean metric in the Dubrovin-Saito standard setting. Up to $n=4$ the prepotentials we get coincide with those associated with constrained KP equations discussed in [20].